On the solvability of a boundary value problem for a quasilinear equation of mixed type with two degeneration lines

Abstract The article investigates the existence of a generalized solution to one boundary value problem for an equation of mixed type with two lines of degeneration in the weighted space of S.L. Sobolev. In proving the existence of a generalized solution, the spaces of functions U(Ω) and V (Ω) are introduced, the spaces H 1 (Ω) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> <mml:mo>*</mml:mo> </mml:msubsup> </mml:mrow> </mml:math> (Ω) are defined as the completion of these spaces of functions, respectively, with respect to the weighted norms, including the functions K( y ) and N( x ). Using an auxiliary boundary value problem for a first order partial differential equation, Kondrashov’s theorem on the compactness of the embedding of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> (Ω) in L 2 (Ω) and Vishik’s lemma, the existence of a solution to the boundary value problem is proved.

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